Stationary Points Research Articles

Curvature-driven pathways interpolating between stationary points: the case of the pure spherical 3-spin model

This paper focuses on characterizing the energy profile along pathways connecting different regions of configuration space in the context of a prototypical glass model, the pure spherical p-spin model with p = 3. The study investigates pairs of stationary points (local minima or rank-1 saddles), analyzing the energy profile along geodesic paths and comparing them with ‘perturbed’ pathways correlated to the landscape curvature. The goal is to assess the extent to which information from the local Hessian matrices around stationary points can identify paths with lower energy barriers. Surprisingly, unlike findings in other systems, the direction of softest local curvature is not a reliable predictor of low-energy paths, except in the case in which the direction of softest curvature corresponds to an isolated mode of the Hessian. However, other information encoded in the local Hessian does allow the identification of pathways associated with lower energy barriers. We conclude commenting on implications for the system’s activated dynamics.

A uniqueness criterion and a counterexample to regularity in an incompressible variational problem

In this paper we consider the problem of minimizing functionals of the form E(u)=∫Bf(x,∇u)dx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E(u)=\\int _B f(x,\ abla u) \\,dx$$\\end{document} in a suitably prepared class of incompressible, planar maps u:B→R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u: B \\rightarrow \\mathbb {R}^2$$\\end{document}. Here, B is the unit disk and f(x,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x,\\xi )$$\\end{document} is quadratic and convex in ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. It is shown that if u is a stationary point of E in a sense that is made clear in the paper, then u is a unique global minimizer of E(u) provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional f(x,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x,\\xi )$$\\end{document}, depending smoothly on ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} but discontinuously on x, whose unique global minimizer is the so-called N-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N-$$\\end{document}covering map, which is Lipschitz but not C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}.

Nonlinear wavepacket dynamics in proximity to a stationary inflection point

Nonlinear wavepacket dynamics in proximity to a stationary inflection point

(Pyridin-2-ylmethyl)triel Derivatives as Masked Frustrated Lewis Pairs. Interactions and CO2 -Sequestration.

The isolated (pyridin-2-ylmethyl)triel derivatives (triel=B, Al and Ga) show an intramolecular N⋅⋅⋅Tr triel bond as shown by compounds found in the Cambridge Structural Database and DFT calculations. The possibility to use them as masked frustrated Lewis pairs (mFLP) has been explored theoretically concerning their reaction with CO2 . The adduct formation proceeds in two steps. In the first one, the (pyridin-2-ylmethyl)triel derivatives break the intramolecular N⋅⋅⋅Tr bond assisted by CO2 and in the second step the adduct is formed with Tr-O and N-C covalent bonds. The corresponding energy minima and transition states (TS) of the reaction have been characterized and analyzed. The distortion/interaction model analysis of the stationary points indicates that the whole process can be divided in two parts: reorganization of the mFLP in the first steps of the reaction while the reaction with CO2 (associated to the distortion of this molecule) is more important in the formation of the final adduct. In all cases studied, the final products are more stable than the starting molecules that combine with reasonable TS energies indicating that these reactions can occur.

Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder.

Cauchy data for Levin’s method

Abstract In this paper, we describe the Cauchy data that gives rise to slowly oscillating solutions to the Levin equation. We present a general result on the existence of a unique minimizer of $\|Bx\|$ subject to the constraint $Ax=y$, where $A,B$ are linear, but not necessarily bounded operators on a complex Hilbert space. This result is used to obtain the solution to the Levin equation, both in the univariate and multivariate case, which minimizes the mean-square of the derivative over the domain. The Cauchy data that generates this solution is then obtained, and this can be used to supplement the Levin equation in the computation of highly oscillatory integrals in the presence of stationary points.

Untrained neural network embedded Fourier phase retrieval from few measurements

Fourier phase retrieval (FPR) is a challenging task used in various applications to recover an unknown signal from its Fourier phaseless measurements. FPR with few measurements helps reduce time and hardware costs but suffers from severe ill-posedness. Recently, untrained neural networks have offered new approaches by introducing learned priors to alleviate ill-posedness without requiring external data. However, they are computationally expensive and not ideal for reconstructing high-frequency structures in images. This paper proposes an untrained neural network (NN) embedded FPR algorithm, based on the alternating direction method of multipliers. Specifically, we use a generative network to represent the image to be recovered, which confines the image to the space defined by the network structure. Additionally, total variation regularization is imposed to facilitate the preservation of image edges and local structures. Furthermore, to reduce the computational cost mainly caused by the updates of NN’s parameters, we develop an accelerated algorithm that adaptively trades off between explicit and implicit regularization. We theoretically analyze that the proposed algorithm can converge to a stationary point of the objective function under mild conditions. Experimental results indicate that the proposed algorithm outperforms existing untrained-NN-based algorithms with fewer computational resources and even performs competitively against trained-NN-based algorithms.

Insights into the Mechanism, Regio-/Diastereoselectivities and Ligand Role of Nickel-Initiated [3+2] Cycloadditions between Vinylcyclopropane and N-Tosylbenzaldimine

Density functional theory (DFT) was employed to explore the reaction mechanism, regio- and diastereoselectivities of nickel-initiated [3+2] cycloaddition between vinylcyclopropane (VCP) and N-tosylbenzaldimine assisted by phosphine ligands. Four different binding modes of the nickel center to VCP substrate were explored during the ring-opening of VCP, among which the C,C_anti and C,C_syn modes were verified to be the most accessible ones. Further explorations about four different phosphine ligand-assisted reactions based on the two most probable binding modes show that the difference in binding mode of bi- and monodentate phosphine ligands can vary the optimal reaction pathway, especially in the [3+2] cycloaddition process between the ring-opened intermediate and N-tosylbenzaldimine. The formation of C–C and C–N bonds between N-tosylbenzaldimine and the ring-opened intermediate through [3+2] cycloaddition is found to be stepwise, with the former acting as the rate-determining step (RDS) in most cases. Computed free energy barriers of RDS transition states on the optimal path I or II not only give out good predictions for reaction rates and half-lives, but also provide reasonable explanations for the major generation of cis-pyrrolidine. Noncovalent interaction analyses of key stationary points suggest the rate is influenced by both electronic effects and steric hindrance, while the diastereoselectivity is mainly controlled by electronic effects.

Numerical Integration of Highly Oscillatory Functions with and without Stationary Points

This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin’s algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active use of Chebyshev interpolation when using the collocation method on Gauss–Lobatto grids, it is possible to achieve high speed and stability when taking into account a large number of collocation points. The presented spectral method of integrating factors is both flexible and reliable and allows for avoiding the ambiguities that arise when applying the classical method of collocation for the ODE solution (Levin) in the physical space. The new method can serve as a basis for solving ordinary differential equations of the first and second orders when creating high-efficiency software, which is demonstrated by solving several model problems.

Electron-beam-promoted fullerene dimerization in nanotubes: insights from DFT computations.

Fullerene dimerization inside a peapod is analyzed at DFT level by characterizing the stationary points and deriving the energy profile of the initial and reversible process named phase 1. We find that the barriers for the radical cation mechanism are significantly lower than those found for the neutral pathway. The peapod is mainly providing one-dimensional confinement for the reaction to take place in a more efficient way. Car-Parrinello metadynamics simulations provide hints on structures for the initial steps of the irreversible phase 2 where bond formation and breaking lead to important structural reorganizations within the coalescence process.

Escaping saddle points efficiently with occupation-time-adapted perturbations

Motivated by the super-diffusivity of self-repelling random walk, which has roots in statistical physics, this paper develops a new perturbation mechanism for optimization algorithms. In this mechanism, perturbations are adapted to the history of states via the notion of occupation time. After integrating this mechanism into the framework of perturbed gradient descent (PGD) and perturbed accelerated gradient descent (PAGD), two new algorithms are proposed: perturbed gradient descent adapted to occupation time (PGDOT) and its accelerated version (PAGDOT). PGDOT and PAGDOT are guaranteed to avoid getting stuck at non-degenerate saddle points, and are shown to converge to second-order stationary points at least as fast as PGD and PAGD, respectively. The theoretical analysis is corroborated by empirical studies in which the new algorithms consistently escape saddle points and outperform not only their counterparts, PGD and PAGD, but also other popular alternatives including stochastic gradient descent, Adam, and several state-of-the-art adaptive gradient methods.

Impact of the T296S mutation in P450 GcoA for aryl-O-demethylation: a QM/MM study.

Lignin, a complex plant cell wall component, holds promise as a renewable aromatic carbon feedstock. p-Vanillin is a key product of lignin depolymerization and a precursor of protocatechuic acid (PCA) that has tremendous potential for biofuel production. While the GcoAB enzyme, native to Amycolatopsis sp., naturally catalyzes aryl-O-demethylation toward guaiacol, recent research introduced a single mutation, T296S, into the GcoAP450 enzyme, enabling it to catalyze aryl-O-demethylation of p-vanillin. This structural modification increases the efficiency of GcoAP450 for the natural substrate while being active for p-vanillin. This study reveals the increased flexibility of p-vanillin and its ability to adapt a favorable conformation by aligning the methoxy group in close proximity to Fe(IV) = O of Cpd I in the active site of the T296S variant. The QM/MM calculations in accordance with the experimental data validated that the rate-limiting step for the oxidation of p-vanillin is hydrogen atom abstraction and provided a detailed geometric structure of stationary and saddle points for the oxidation of p-vanillin.

Exploring the Tl H potential energy surface: A comparative analysis with group 13 systems and experiment.

Thallium chemistry is experiencing unprecedented importance. Therefore, it is valuable to characterize some of the simplest thallium compounds. Stationary points along the singlet and triplet Tl H potential energy surface have been characterized. Stationary point geometries were optimized with the CCSD(T)/aug-cc-pwCVQZ-PP method. Harmonic vibrational frequencies were computed at the same level of theory while anharmonic vibrational frequencies were computed at the CCSD(T)/aug-cc-pwCVTZ-PP level of theory. Final energetics were obtained with the CCSDT(Q) method. Basis sets up to augmented quintuple-zeta cardinality (aug-cc-pwCV5Z-PP) were employed to obtain energetics in order to extrapolate to the complete basis set limits using the focal point approach. Zero-point vibrational energy corrections were appended to the extrapolated energies in order to determine relative energies at 0 K. It was found that the planar dibridged isomer lies lowest in energy while the linear structure lies highest in energy. The results were compared to other group 13 M H (M = B, Al, Ga, In, and Tl) theoretical studies and some interesting variations are found. With respect to experiment, incompatibilities exist.

Inexact Exponential Penalty Function with the Augmented Lagrangian for Multiobjective Optimization Algorithms

This paper uses an augmented Lagrangian method based on an inexact exponential penalty function to solve constrained multiobjective optimization problems. Two algorithms have been proposed in this study. The first algorithm uses a projected gradient, while the second uses the steepest descent method. By these algorithms, we have been able to generate a set of nondominated points that approximate the Pareto optimal solutions of the initial problem. Some proofs of theoretical convergence are also proposed for two different criteria for the set of generated stationary Pareto points. In addition, we compared our method with the NSGA-II and augmented the Lagrangian cone method on some test problems from the literature. A numerical analysis of the obtained solutions indicates that our method is competitive with regard to the test problems used for the comparison.

Pengaruh Filler Plastik dan Optimasi Faktor Proses Produksi Cellular Lightweight Concrete

The use of plastic has become a common thing in different lives. The effect of using these plastic products is to create plastic waste. As the population increases, the accumulation of plastic waste will increase. So that it can be done by applying the 3R method, namely reduce, reuse, and recycle. Plastic waste is used as a lightweight brick reinforcement material. This study aimed to determine the factors that most influence the density value of cellular lightweight concrete with plastic filler and optimize the production process. The method used is the Response Surface Methodology (RSM). The experimental design was chosen Central Composite Design with three factors. The independent variables used in this study were the amount of plastic, the fraction of sand: cement, and the volume of water. The amount of plastic used is 2 grams, 3 grams, and 4 grams. The fraction of sand:cement was 4:3, 3.3:2, and 2:1. The volume of water used was 60 mL, 80 mL, and 100 mL. The results of this study indicate that the optimal value for the density of lightweight bricks is found in the amount of plastic of 7.745 grams, the fraction of sand:cement is 5.8:2, and the volume of water is 139.509 mL. The surface response from the second-order equation is the saddle point with the optimal point at the stationary point. The stationary points obtained are 2.821, 2.726, and 1.769. The research analysis results show that the independent variable that most influence the density value is the fraction of sand: cement.

Global Optimality Guarantees for Policy Gradient Methods

Policy gradient methods, which have powered a lot of recent success in reinforcement learning, search for an optimal policy in a parameterized policy class by performing stochastic gradient descent on the cumulative expected cost-to-go under some initial state distribution. Although widely used, these methods lack theoretical guarantees as the optimization objective is typically nonconvex even for simple control problems, and hence are understood to only converge to a stationary point. In “Global Optimality Guarantees for Policy Gradient Methods,” J. Bhandari and D. Russo identify structural properties of the underlying MDP that guarantee that despite nonconvexity, the optimization objective has no suboptimal stationary points, ensuring asymptotic convergence of policy gradient methods to globally optimal policies. Under stronger conditions, authors show the policy gradient objective to satisfy a Polyak-lojasiewicz (gradient dominance) condition that yields fast convergence rates. In addition, when some of the said conditions are relaxed, authors provide bounds on the suboptimality gap of any stationary point. The results rely on a key connection with policy iteration, a classic dynamic programming algorithm which solves a single period optimization problem at every step. The authors show how structure in the single period optimization problems solved by policy iteration translate into nice properties of the multiperiod policy gradient objective, making it amenable for first-order methods to find globally optimal solutions. The authors also instantiate their framework for several classical control problems including tabular and linear MDPs, linear quadratic control, optimal stopping, and finite-horizon inventor control problems.

Estimation of sparse covariance matrix via non-convex regularization

Estimation of high-dimensional sparse covariance matrix is one of the fundamental and important problems in multivariate analysis and has a wide range of applications in many fields. This paper presents a novel method for sparse covariance matrix estimation via solving a non-convex regularization optimization problem. We establish the asymptotic properties of the proposed estimator and develop a multi-stage convex relaxation method to find an effective estimator. The multi-stage convex relaxation method guarantees any accumulation point of the sequence generated is a first-order stationary point of the non-convex optimization. Moreover, the error bounds of the first two stage estimators of the multi-stage convex relaxation method are derived under some regularity conditions. The numerical results show that our estimator outperforms the state-of-the-art estimators and has a high degree of sparsity on the premise of its effectiveness.

Gas-Phase Phenyl Radical + O2 Reacts via a Submerged Transition State.

In the gas-phase chemistry of the atmosphere and automotive fuel combustion, peroxyl radical intermediates are formed following O2 addition to carbon-centered radicals which then initiate a complex network of radical reactions that govern the oxidative processing of hydrocarbons. The rapid association of the phenyl radical-a fundamental radical related to benzene-with O2 has hitherto been modeled as a barrierless process, a common assumption for peroxyl radical formation. Here, we provide an alternate explanation for the kinetics of this reaction by deploying double-hybrid density functional theory (DFT), at the DSD-PBEP86-D3(BJ)/aug-cc-pVTZ level of theory, and locate a submerged adiabatic transition state connected to a prereaction complex along the reaction entrance pathway. Using this potential energy scheme, experimental rate coefficients k(T) for the addition of O2 to the phenyl radical are accurately reproduced within a microcanonical kinetic model. This work highlights that purportedly barrierless radical oxidation reactions may instead be modeled using stationary points, which in turn provides insight into pressure and temperature dependence.

Arrangement of nearby minima and saddles in the mixed spherical energy landscapes

The mixed spherical models were recently found to violate long-held assumptions about mean-field glassy dynamics. In particular, the threshold energy, where most stationary points are marginal and that in the simpler pure models attracts long-time dynamics, seems to lose significance. Here, we compute the typical distribution of stationary points relative to each other in mixed models with a replica symmetric complexity. We examine the stability of nearby points, accounting for the presence of an isolated eigenvalue in their spectrum due to their proximity. Despite finding rich structure not present in the pure models, we find nothing that distinguishes the points that do attract the dynamics. Instead, we find new geometric significance of the old threshold energy, and invalidate pictures of the arrangement of most marginal inherent states into a continuous manifold.

Cn, CnH and their anions: Quest for linearity with n≤8 even versus odd, and beyond

AbstractUsing the concept of quasi‐molecule (“tile”) and the database of quasi‐molecules embedded on a parent molecule, it is discussed whether the latter can attain linear form or otherwise. Besides anew accurate optimization of all tiles (quasi‐triatomics) at various levels of ab initio theory and basis‐sets, the nature of the predicted stationary points for the title parent molecules is probed through a priori calculations here too reported. Also discussed is the common rule that even‐ anions are linear while odd‐numbered ones tend to have nonlinear isomers. The reported quasi‐molecule approach is general, and allow the prediction of linearity or otherwise of the parent systems prior to calculations on them. When based on an extension of the bisection method (Varandas, Int. J. Quantum Chem. 2023, 123, e27036.), it is easy to use even for large parent molecules, as illustrated for neutral and anionic carbon clusters with .